\(\int \sin ^5(e+f x) (a+b \sin ^2(e+f x))^p \, dx\) [172]

Optimal result

Integrand size = 23, antiderivative size = 220 \[ \int \sin ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\frac {(3 a-2 b (2+p)) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{1+p}}{b^2 f (3+2 p) (5+2 p)}-\frac {\left (3 a^2-4 a b (1+p)+4 b^2 \left (2+3 p+p^2\right )\right ) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {b \cos ^2(e+f x)}{a+b}\right )}{b^2 f (3+2 p) (5+2 p)}-\frac {\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{1+p} \sin ^2(e+f x)}{b f (5+2 p)} \]

[Out]

(3*a-2*b*(2+p))*cos(f*x+e)*(a+b-b*cos(f*x+e)^2)^(p+1)/b^2/f/(4*p^2+16*p+15)-(3*a^2-4*a*b*(p+1)+4*b^2*(p^2+3*p+
2))*cos(f*x+e)*(a+b-b*cos(f*x+e)^2)^p*hypergeom([1/2, -p],[3/2],b*cos(f*x+e)^2/(a+b))/b^2/f/(4*p^2+16*p+15)/((
1-b*cos(f*x+e)^2/(a+b))^p)-cos(f*x+e)*(a+b-b*cos(f*x+e)^2)^(p+1)*sin(f*x+e)^2/b/f/(5+2*p)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3265, 427, 396, 252, 251} \[ \int \sin ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=-\frac {\left (3 a^2-4 a b (p+1)+4 b^2 \left (p^2+3 p+2\right )\right ) \cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {b \cos ^2(e+f x)}{a+b}\right )}{b^2 f (2 p+3) (2 p+5)}+\frac {(3 a-2 b (p+2)) \cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{p+1}}{b^2 f (2 p+3) (2 p+5)}-\frac {\sin ^2(e+f x) \cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{p+1}}{b f (2 p+5)} \]

[In]

Int[Sin[e + f*x]^5*(a + b*Sin[e + f*x]^2)^p,x]

[Out]

((3*a - 2*b*(2 + p))*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(1 + p))/(b^2*f*(3 + 2*p)*(5 + 2*p)) - ((3*a^2 -
4*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/
2, (b*Cos[e + f*x]^2)/(a + b)])/(b^2*f*(3 + 2*p)*(5 + 2*p)*(1 - (b*Cos[e + f*x]^2)/(a + b))^p) - (Cos[e + f*x]
*(a + b - b*Cos[e + f*x]^2)^(1 + p)*Sin[e + f*x]^2)/(b*f*(5 + 2*p))

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 3265

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \left (1-x^2\right )^2 \left (a+b-b x^2\right )^p \, dx,x,\cos (e+f x)\right )}{f} \\ & = -\frac {\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{1+p} \sin ^2(e+f x)}{b f (5+2 p)}+\frac {\text {Subst}\left (\int \left (a+b-b x^2\right )^p \left (a-2 b (2+p)-(3 a-2 b (2+p)) x^2\right ) \, dx,x,\cos (e+f x)\right )}{b f (5+2 p)} \\ & = \frac {(3 a-2 b (2+p)) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{1+p}}{b^2 f (3+2 p) (5+2 p)}-\frac {\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{1+p} \sin ^2(e+f x)}{b f (5+2 p)}-\frac {\left (3 a^2-4 a b (1+p)+4 b^2 \left (2+3 p+p^2\right )\right ) \text {Subst}\left (\int \left (a+b-b x^2\right )^p \, dx,x,\cos (e+f x)\right )}{b^2 f (3+2 p) (5+2 p)} \\ & = \frac {(3 a-2 b (2+p)) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{1+p}}{b^2 f (3+2 p) (5+2 p)}-\frac {\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{1+p} \sin ^2(e+f x)}{b f (5+2 p)}-\frac {\left (\left (3 a^2-4 a b (1+p)+4 b^2 \left (2+3 p+p^2\right )\right ) \left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p}\right ) \text {Subst}\left (\int \left (1-\frac {b x^2}{a+b}\right )^p \, dx,x,\cos (e+f x)\right )}{b^2 f (3+2 p) (5+2 p)} \\ & = \frac {(3 a-2 b (2+p)) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{1+p}}{b^2 f (3+2 p) (5+2 p)}-\frac {\left (3 a^2-4 a b (1+p)+4 b^2 \left (2+3 p+p^2\right )\right ) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {b \cos ^2(e+f x)}{a+b}\right )}{b^2 f (3+2 p) (5+2 p)}-\frac {\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{1+p} \sin ^2(e+f x)}{b f (5+2 p)} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.58 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.45 \[ \int \sin ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (3,\frac {1}{2},-p,4,\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right ) \sqrt {\cos ^2(e+f x)} \sin ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {a+b \sin ^2(e+f x)}{a}\right )^{-p} \tan (e+f x)}{6 f} \]

[In]

Integrate[Sin[e + f*x]^5*(a + b*Sin[e + f*x]^2)^p,x]

[Out]

(AppellF1[3, 1/2, -p, 4, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*Sin[e + f*x]^5*(a + b*S
in[e + f*x]^2)^p*Tan[e + f*x])/(6*f*((a + b*Sin[e + f*x]^2)/a)^p)

Maple [F]

\[\int \left (\sin ^{5}\left (f x +e \right )\right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{p}d x\]

[In]

int(sin(f*x+e)^5*(a+b*sin(f*x+e)^2)^p,x)

[Out]

int(sin(f*x+e)^5*(a+b*sin(f*x+e)^2)^p,x)

Fricas [F]

\[ \int \sin ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right )^{5} \,d x } \]

[In]

integrate(sin(f*x+e)^5*(a+b*sin(f*x+e)^2)^p,x, algorithm="fricas")

[Out]

integral((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*(-b*cos(f*x + e)^2 + a + b)^p*sin(f*x + e), x)

Sympy [F(-1)]

Timed out. \[ \int \sin ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\text {Timed out} \]

[In]

integrate(sin(f*x+e)**5*(a+b*sin(f*x+e)**2)**p,x)

[Out]

Timed out

Maxima [F]

\[ \int \sin ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right )^{5} \,d x } \]

[In]

integrate(sin(f*x+e)^5*(a+b*sin(f*x+e)^2)^p,x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e)^2 + a)^p*sin(f*x + e)^5, x)

Giac [F]

\[ \int \sin ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right )^{5} \,d x } \]

[In]

integrate(sin(f*x+e)^5*(a+b*sin(f*x+e)^2)^p,x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e)^2 + a)^p*sin(f*x + e)^5, x)

Mupad [F(-1)]

Timed out. \[ \int \sin ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int {\sin \left (e+f\,x\right )}^5\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p \,d x \]

[In]

int(sin(e + f*x)^5*(a + b*sin(e + f*x)^2)^p,x)

[Out]

int(sin(e + f*x)^5*(a + b*sin(e + f*x)^2)^p, x)